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%I #8 Jun 29 2020 22:21:45
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T 0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0
%N Number of (2,1,2)-matching permutations of the prime indices of n.
%C Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%F a(n) + A335450(n) = A008480(n).
%e The a(n) permutations for n = 18, 36, 54, 72, 90, 108, 144, 180:
%e (212) (1212) (2122) (11212) (2123) (12122) (111212) (12123)
%e (2112) (2212) (12112) (2132) (12212) (112112) (12132)
%e (2121) (12121) (2312) (21122) (112121) (12312)
%e (21112) (3212) (21212) (121112) (13212)
%e (21121) (21221) (121121) (21123)
%e (21211) (22112) (121211) (21132)
%e (22121) (211112) (21213)
%e (211121) (21231)
%e (211211) (21312)
%e (212111) (21321)
%e (23112)
%e (23121)
%e (31212)
%e (32112)
%e (32121)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x>y]&]],{n,100}]
%Y References found in the link are not all repeated here.
%Y Positions of ones are A095990.
%Y The avoiding version is A335450.
%Y Replacing (2,1,2) with (1,2,1) gives A335446.
%Y Patterns are counted by A000670.
%Y Permutations of prime indices are counted by A008480.
%Y Unsorted prime signature is A124010. Sorted prime signature is A118914.
%Y (1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
%Y STC-numbers of permutations of prime indices are A333221.
%Y (1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.
%Y Patterns matched by standard compositions are counted by A335454.
%Y (1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
%Y (1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
%Y Dimensions of downsets of standard compositions are A335465.
%Y (1,2,2)-matching compositions are ranked by A335475.
%Y (2,2,1)-matching compositions are ranked by A335477.
%Y Cf. A056239, A056986, A112798, A158005, A181796, A335452, A335463.
%K nonn
%O 1,36
%A _Gus Wiseman_, Jun 14 2020
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